Does washing the lips with water and soap during the day in Ramadan—while being careful not to swallow anything—affect the validity of the fast if something from that inadvertently enters the mouth?

Your fast is valid, and there is nothing against you. The default is that the fast is valid, and it is not invalidated merely by doubting its invalidity. Even if you accidentally swallowed some water and soap unintentionally, your fast is valid. You must beware of obsessive thoughts and turn away from them.